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Fast Property Testing and Metrics for Permutations

Part of: Extremal combinatorics Discrete mathematics in relation to computer science Algorithms - Computer Science Combinatorics

Published online by Cambridge University Press:  24 May 2018

JACOB FOX  and
FAN WEI
JACOB FOX
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail: jacobfox@stanford.edu, fanwei@stanford.edu)
FAN WEI
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail: jacobfox@stanford.edu, fanwei@stanford.edu)
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Abstract

The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are ε-far from satisfying the property. There are now several general results in this area which show that natural properties of combinatorial objects can be tested with ‘constant’ query complexity, depending only on ε and the property, and not on the size of the object being tested. The upper bound on the query complexity coming from the proof techniques is often enormous and impractical. It remains a major open problem if better bounds hold.

Maybe surprisingly, for testing with respect to the rectangular distance, we prove there is a universal (not depending on the property), polynomial in 1/ε query complexity bound for two-sided testing hereditary properties of sufficiently large permutations. We further give a nearly linear bound with respect to a closely related metric which also depends on the smallest forbidden subpermutation for the property. Finally, we show that several different permutation metrics of interest are related to the rectangular distance, yielding similar results for testing with respect to these metrics.

MSC classification

Primary: 68W20: Randomized algorithms
Secondary: 68R05: Combinatorics 05D40: Probabilistic methods 05A05: Permutations, words, matrices 68R15: Combinatorics on words

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Special Issue 4: Special Issue: Oberwolfach Workshop , July 2018 , pp. 539 - 579
Copyright
Copyright © Cambridge University Press 2018 

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